natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Let $G$ be an abelian group, and $n$ a natural number. The Eilenberg-MacLane space $K(G,n)$ is the unique space that has $\pi_n(K(G,n))\cong G$ and its other homotopy groups trivial. These spaces are a basic tool in classical algebraic topology, they can be used to define cohomology.
There is an analogous construction in homotopy type theory.
Let $G$ be a group. The Eilenberg-MacLane space type $K(G,1)$ is the higher inductive 1-type with the following constructors:
$base$ is a point of $K(G,1)$, $loop$ is a function that constructs a path from $base$ to $base$ for each element of $G$. $id$ says the path constructed from the identity element is the trivial loop. Finally, $comp$ says that the path constructed from group multiplication of elements is the concatenation of paths.
It should be noted that this type is 1-truncated which could be added as another constructor:
To define a function $f : K(G,1) \to C$ for some type $C$, it suffices to give
Then $f$ satisfies the equations
It should be noted that the above can be compressed, to specify a function $f : K(G,1) \to C$, it suffices to give:
Dan Licata, Eric Finster, Eilenberg-MacLane spaces in homotopy type theory, CSL-LICS ‘14 66 (2014) 1-9 [pdf text, Agda HoTT code, web discussion, doi:10.1145/2603088.2603153]
Floris van Doorn, §42 in: On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory (2018) [arXiv:1808.10690]
Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types [arXiv:2301.02636]
David Wärn, Eilenberg-MacLane spaces and stabilisation in homotopy type theory, J. Homotopy Relat. Struct. 18 (2023) 357–368 [arXiv:2301.03685, doi:10.1007/s40062-023-00330-5]
Formal proof that the $K(G,-)$ form a spectrum:
Last revised on January 31, 2024 at 09:28:09. See the history of this page for a list of all contributions to it.